Vertex quasiprimitive two-geodesic transitive graphs

Abstract

For a non-complete graph $\Gamma$, a vertex triple $(u,v,w)$ with $v$ adjacent to both $u$ and $w$ is called a $2$-geodesic if $u\neq w$ and $u,w$ are not adjacent. Then $\Gamma$ is said to be $2$-geodesic transitive if its automorphism group is transitive on both arcs and 2-geodesics. In previous work the author showed that if a $2$-geodesic transitive graph $\Gamma$ is locally disconnected and its automorphism group $\Aut(\Gamma)$ has a non-trivial normal subgroup which is intransitive on the vertex set of $\Gamma$, then $\Gamma$ is a cover of a smaller 2-geodesic transitive graph. Thus the `basic' graphs to study are those for which $\Aut(\Gamma)$ acts quasiprimitively on the vertex set. In this paper, we study 2-geodesic transitive graphs which are locally disconnected and $\Aut(\Gamma)$ acts quasiprimitively on the vertex set. We first determine all the possible quasiprimitive action types and give examples for them, and then classify the family of $2$-geodesic transitive graphs whose automorphism group is primitive on its vertex set of $\PA$ type.